AN - How to create path for moving an object or an outline?
Thanks for posting this question.  The animation you found is very cool, and a fun practice project with animation nodes.



I made this mockup with just two circles, but the method can easily be extended to as many circles as desired.  Here are the AN nodes, in detail.



The animation is built around the large central circle centered at point A (0,0), which is a Bezier curve.  Since the circle is a bezier curve, you can use the AN Evaluate Spline node and find any point along the curve.



Using the Timeline Time Info node, animate location B along the curve using the Timeline Info Frame # / 360.  Using Modulo gives just the remainder, so it will repeat after 360 frames.  

Using the center point of the circle and the animated point on the perimeter, draw radius line AB.



Use the Object Instancer to create a copy of the circle.  Using Object Transforms Output, rescale the second circle to 1/3 of the original size, and change location to point B.   As before, animate point C along the smaller circle, but this time using Frame/120 since it rotates 3x as fast.  Draw radius line BC as before. 



To draw the curve, draw a horizontal line from C to D.  C(x,y) moves with the circles, and D moves vertically only.  D(x) is fixed, D(y) = C(y).

As D moves, it creates the curve E.  A very small plane is set up as a particle emitter, and its location is updated to follow point D.  Over time, emits particles which move to the right at a steady rate of speed.  

By joining these particles into a spline using Spline from Points node, then smoothing it a bit, a nice curve emerges.



Here are shows the emitter settings I used.  Note that gravity is set to 0, and the particle velocity has a positive velocity on the Normal, dimension, so it emits directly out of its face, but 0 otherwise.  I've rotated the emitter so that it faces to the right (+X) in the diagram, so it will emit from it's face, in the +X direction.  



Here is the .blend file.   It was created in Blender 2.79, using Animation Nodes 2.0.4.  I don't know if it's compatible with the more recent Blender or AN, but I hope you could reproduce this animation if needed.

HTH

-Z